The topic of approximate conservation has a broad interest and has been related to approximate
symmetry, widely studied in classical and quantum mechanics. Referring to the broad range of phenomena in
physics, from nuclear forces to gravitation, having near symmetry, R. Feynman concludes that
“God made the laws only nearly symmetrical so that we should not be jealous of His perfection” [1]}.
Approximate continuous symmetries (Lie-Bäcklund symmetries) were studied for differential equations
with
or without a Lagrangian (see [2]} and [3]}, respectively). Beyond their utility in the theory of
regular and singular perturbations,
these symmetries can be used to gain insight into the dynamics of complex systems.
Approximate conservation laws, valid for certain concentrations and not valid for other concentrations
of biochemical species, imply that the same biochemical system can have multiple behaviors depending
on the internal or external stimuli.
In biology, “imperfect” conservation allows living systems to be
flexible, to evolve and adapt to changes of the environment.
This also leads to multiple dynamical phenomena: slow metastable states,
bifurcations in the fast dynamics of the system and itineracy when the system
switches from one metastable state to another [4]}, [5]}, [6]}.
